Abstract
The formulation of optimal control problems governed by Cauchy-Riemann equations is presented. A distributed control mechanism through divergence and curl sources is considered with the boundary conditions of mixed type. A Lagrange multiplier framework is introduced to characterize the solution to Cauchy-Riemann optimal control problems as the solution of an optimality system of four first-order partial differential equations and two optimality conditions. To solve the optimality system, staggered grids and multigrid methods are investigated. It results that staggered grids provide a natural collocation of the optimization variables and second-order accurate solutions are obtained. The proposed multigrid scheme is based on a coarsening by a factor of three that results in a nested hierarchy of staggered grids. On these grids a distributed-Gauss-Seidel and gradient-based smoothing scheme is employed. Results of numerical experiments validate the proposed optimal control formulation and demonstrate the effectiveness of the staggered-grids multigrid solution procedure.
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Communicated by C. W. Oosterlee and A. Borzi.
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Butt, M.M., Borzì, A. Formulation and multigrid solution of Cauchy-Riemann optimal control problems. Comput. Visual Sci. 14, 79–90 (2011). https://doi.org/10.1007/s00791-011-0161-9
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DOI: https://doi.org/10.1007/s00791-011-0161-9